In an attempt to uncover Spinoza’s logic, I’m going to transcribe his propositions without the exact verbiage of his demonstrations, etc., and provide my own words for understanding what he means (or at least reword his demonstrations for myself).

P1: A substance is prior to its affections

All affections (modes, modifications, expressions) are necessarily of some original substance. D3: a substance “is in itself and is conceived through itself” and per D5, a mode is defined as “affections of a substance, or that which is in another through which it is also conceived.” I think D1 also applies here since a substance is causa sui…

P2: Two substances having different attributes have nothing in common with one another.

Per D4, an attribute is what the intellect perceives of a substance as constituting its essence. This statement in itself appears to be significant because we are only capable of perceiving attributes, rather than substances directly. If attributes convey a substance’s essence, then differing attributes belong to different substances. In Descartes’s Principles of Philosophy, he considers the body and mind to be two distinct created substances, and asserts that these substances’ principal attributes are extension and thought, respectively. At this point Spinoza does not appear to be challenging Descartes.

P3: If things have nothing in common with one another, one of them cannot be the cause of the other.

Per A5: “Things that have nothing in common with one another also cannot be understood through one another…” and A4: “The knowledge of an effect depends on…the knowledge of its cause,” his demonstration shows that two distinct things cannot be understood through one another (understanding wood does not make water more intelligible) and that one can not be the cause of the other.

P4: Two or more distinct things are distinguished from one another, either by a difference in the attributes of the substances or by a difference in their affections.

Everything is, per A1, “either in itself or in another.” Using the definitions of substance (in itself) and affection (is through another) referenced in P1. Spinoza tells us explicitly in this demonstration that “outside the intellect there is nothing except substances and their affections” but I am wondering how the intellect came to be neither substance nor affection. This appears to be answered by the wording of his definitions (D3 and D5) where he says that both substances and affections are “conceived” (either through themselves or through others, respectively). Therefore, there must be an intellect doing this conceiving. These categories – substance and affection – thereby constitute everything; nothing else exists by which to distinguish between two things. Regarding the wording of this proposition, we must remember that, per D4, attributes are what we perceive of a substance and consider to be its essence. As humans, it appears that we cannot directly perceive or experience a substance.

P5: In Nature there cannot be two or more substances of the same nature or attribute.

This proposition appears to inaugurate the challenge to Descartes. Assuming there could be multiple substances, they “would have to be distinguished from one another either by a difference in their attributes, or by a difference in their affections,” according to P4. Again, this is because all that exists in the universe is either substance (with its attendant attributes) or affection (expression(s) of a substance). From this point, we can recall Spinoza’s assertion that any attribute can belong to only one substance. So any two (or more) things with shared attributes are constituted by the same substance. Furthermore, if affections are understood as expressions of a substance, then we can compare the underlying substances, and realize that any two (or more) things with the same affection are again constituted by the same underlying substance.

At this point, Spinoza has not inquired into how substances are created nor has he provided any information about how they might affect one another. Rather, he has just – I say “just,” as if this were some minor accomplishment! – presented how we should understand a substance and how to compare it to another. However, based on P5, we can clearly see how he is staking out the idea of “substance monism”: any repetition of shared attributes or affections refers to the same originary substance.